What's the meaning of tangent in the velocity panel in the particle settings?
Strictly, 'tangent' means a vector that touches the surface at precisely one point. Not the surface itself. So in the particle setting context:

tangent controls the starting force that is applied to the particle parallel to the emitter surface.

Normal controls the force that is applied to the particle perpendicular to the emitter surface.

To put it another way,

tangent = 1, normal = 0 will make the particle emit 'sideways'.

tangent = 0, normal = 1 will make the particle emit 'out'.

tangent = 1, normal = 1 will make the particle emit at 45degrees to the surface.